Scaling Laws of Polyelectrolyte Adsorption

Scaling Laws of Polyelectrolyte Adsorption

Itamar Borukhov and David Andelman*

School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv, Israel

Henri Orland

Service de Physique Theorique, CE-Saclay, 91191 Gif-sur-Yvette Cedex, France

Received May 27, 1997; Revised Manuscript Received October 28, 1997

ABSTRACT: Adsorption of charged polymers (polyelectrolytes) from a semidilute solution to a chargedsurface is investigated theoretically. We obtain simple scaling laws for (i) the amount of polymer Γadsorbed to the surface and (ii) the width D of the adsorbed layer, as a function of the fractional chargeper monomer p and the salt concentration cb. For strongly charged polyelectrolytes (p j 1) in a low-saltsolution, both Γ and D scale as p-1/2. In high-salt solutions D ∼ cb

1/2/p whereas the scaling behavior of Γdepends on the strength of the polymer charge. For weak polyelectrolytes (p , 1) we find thatΓ ∼ p/cb

1/2, and for strong polyelectrolytes Γ ∼ cb1/2/p. Our results are in good agreement with adsorption

experiments and with numerical solutions of mean-field equations.

I. IntroductionPolyelectrolytes (charged polymers) are widely used

in industrial applications. For example, many colloidalsuspensions can be stabilized by the adsorption ofpolyelectrolytes. In many experiments, the total amountof polymer adsorbed on a surface (the polymer surfaceexcess) is measured as a function of the bulk polymerconcentration, pH and/or ionic strength of the bulksolution.1-8 (For reviews see, e.g., refs. 9-12). Morerecently, spectroscopy3 and ellipsometry7 have beenused to measure the width of the adsorbed polyelectro-lyte layer. Other techniques such as neutron scatteringcan be employed to measure the entire profile of theadsorbed layer.13,14The theoretical treatment of polyelectrolytes in solu-

tion is not very well established because of the delicateinterplay between the chain connectivity and the longrange nature of electrostatic interactions.15-18 In manystudies adsorption of polyelectrolytes is treated as anextension of neutral polymer theories. In these ap-proaches the polymer concentration profile is deter-mined by minimizing the overall free energy.One approach is a discretemulti-Stern layermodel,19-23

in which the system is placed on a lattice whose sitescan be occupied by a monomer, a solvent molecule, or asmall ion. The electrostatic potential is determined self-consistently together with the concentration profiles ofthe polymer and the small ions. Another approachtreats the electrostatic potential and the polyelectrolyteconcentration as continuous functions.24-28 These quan-tities are obtained from two coupled differential equa-tions derived from the total free energy of the system.In the present work we focus on the adsorption

behavior of polyelectrolytes near a single chargedsurface held at a constant potential. Simple scalingexpressions are presented and compared to concentra-tion profiles that we obtain from exact numericalsolutions, and to experiments measuring the amountof polymer adsorbed on the surface. In section I theadsorption problem is treated numerically. We thenpresent in section II simple scaling arguments describ-ing the adsorption characteristics and, in section III, we

compare our scaling results to experiments. Finally, wepresent our conclusions and some future prospects.

II. Numerical ProfilesConsider a semi-dilute solution of polyelectrolytes in

good solvent in contact with a charged surface (Figure1). In addition to the polymer chains and their coun-terions, the solution contains small ions (salt) assumedhereafter to be monovalent. The system is coupled to abulk reservoir containing polyelectrolyte chains and salt.In the present work we assume that the charge densityon the polymer chains is continuous and uniformlydistributed along the chains. This assumption is validas long as the electrostatic potential is not too high,|âeψ| < 1, where 1/â ) kBT is the thermal energy, e isthe electron charge, and ψ is the electrostatic potential.Further treatments of the polymer charge distribution(annealed and quenched models) can be found in refs.27 and 28.Within mean-field approximation, the free energy of

the system can be expressed in terms of the localelectrostatic potential ψ(r), the local monomer concen-tration c(r) and the local concentration of positive andnegative ions c((r). It is convenient to introduce thepolymer order parameter φ(r) where c(r) ) |φ(r)|2. Theexcess free energy with respect to the bulk is then25-28

The polymer contribution is

where the first term is the polymer elastic energy, abeing the effective monomer size. The second term isthe excluded volume contribution where v is of ordera3. The last term couples the system to the reservoir,where µp is the chemical potential of the polymers andc(∞) ) φb

2 is the bulk monomer concentration.

F ) ∫ dr {fpol(r)+ fions(r)+ fel(r)} (1)

fpol(r) ) kBT[ a26 |∇φ|2 + 12

υ(φ4 - φb4)] - µp (φ

2 - φb2)(2)

1665Macromolecules 1998, 31, 1665-1671

S0024-9297(97)00730-4 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 02/13/1998

The entropic contribution of the small (monovalent)ions is

where ci(r), cbi , and µi are, respectively, the local con-

centration, the bulk concentration, and the chemicalpotential of the i ) ( ions. Finally, the electrostaticcontributions are

The first three terms are the electrostatic energies ofthe monomers, the positive ions, and the negative ions,respectively, p being the fractional charge carried by onemonomer. The last term is the self energy of the electricfield where ε is the dielectric constant of the solution.Note that the electrostatic contribution, eq 4 isequivalent to the well known result: Fel )(ε/8π) ∫ dr|∇ψ|2 plus surface terms. This can be seenby substituting the PoissonsBoltzmann equation (asobtained below) into eq 4 and then integrating by parts.Minimization of the free energy with respect to c(, φ,

and ψ yields a Boltzmann distribution for the densityof the small ions, c((r) ) cb

( exp(-âeψ), and twocoupled differential equations for φ and ψ:

Equation 5 is a generalized PoissonsBoltzmann equa-tion including the free ions as well as the chargedpolymers. The first term represents the salt contribu-tion and the second term is due to the charged mono-mers and their counterions. Equation 6 is a generali-zation of the self-consistent field equation of neutralpolymers.16 In the bulk, the above equations aresatisfied by setting ψ f 0 and φ f φb.When a polyelectrolyte solution is in contact with a

charged surface, the chains adsorb to (or deplete from)the surface, depending on the nature of the monomerssurface interactions. The large number of monomers

on each polymer chain enhances these interactions. Forsimplicity, we assume that the surface is ideal, i.e., flatand homogeneous. In this case physical quantitiesdepend only on the distance x from the surface (seeFigure 1). The surface imposes boundary conditions onthe polymer order parameter φ(x) and electrostaticpotential ψ(x). In thermodynamic equilibrium all chargecarriers in solution should exactly balance the surfacecharges (charge neutrality). The PoissonsBoltzmannequation (eq 5), the self-consistent field equation (eq 6)and the boundary conditions uniquely determine thepolymer concentration profile and the electrostaticpotential. In most cases, these two coupled nonlinearequations can only be solved numerically.In the present work we have chosen the surface to be

at a constant potential ψS, leading to the followingelectrostatic boundary condition

The boundary conditions for φ(x) depend on the natureof the short range interaction of the monomers and thesurface. For simplicity, we take a nonadsorbing surfaceand require that the monomer concentration will vanishthere:

The choice of boundary conditions depends on thephysical system considered. From the numerical pointof view other boundary conditions which include thefirst derivatives of ψ and φ can be implemented as well.Possible variations of these boundary conditions includesurfaces with a constant surface charge29 and surfaceswith a nonelectrostatic short range attractive (or repul-sive) interaction with the polymer.Far from the surface (x f ∞) both ψ and φ reach their

bulk values and their derivatives vanish: ψ′|xf∞ ) 0 andφ′|xf∞ ) 0.The numerical solutions of the mean-field equations

(eqs 5 and 6) together with the boundary conditionsdiscussed above are obtained using a minimal squaremethod. In this method, the spatial coordinate x isdiscretized up to a certain distance far enough from thesurface. An error functional is then calculated bysumming up the squares of the errors in the two mean-field equations at each mesh point. This functional isthen minimized under the constraint that the boundaryconditions are satisfied.In Figure 2 several adsorption profiles calculated

numerically are plotted. The polymer is positivelycharged and is attracted to the nonadsorbing surfaceheld at a constant negative potential. The aqueoussolution contains a small amount of monovalent salt (cb )0.1 mM). The reduced concentration profile c(x)/φb

2 isplotted as a function of the distance from the surface x.Different curves correspond to different values of thereduced surface potential yS ) âeψS, the charge fractionp, and the monomer size a. Although the spatialvariation of the profiles differs in detail, they all havea single peak which can be characterized by its heightand width. We use this feature in the next section toobtain simple analytical expressions characterizing theadsorption.

III. Scaling ResultsThe difficulty in obtaining a simple picture of poly-

electrolyte adsorption lies in the existence of several

Figure 1. Schematic view of a polyelectrolyte solution incontact with a flat surface at x ) 0. The solution containspolyelectrolyte chains and small ions. In our model, the surfaceis held at a constant potential.

fions(r) )

∑i)(

kBT[ci ln ci - ci - cb

i ln cbi + cb

i ] - µi(ci - cbi ) (3)

fel(r) ) peφ2ψ + ec+ψ - ec-ψ - ε

8π|∇ψ|2 (4)

∇2ψ(r) ) 8πeε

cb sinh(âeψ) - 4πeε

(pφ2 - pφb2eâeψ) (5)

a2

6vφ(r) ) ν(φ3 - φb

2φ) + pφâeψ (6)

ψ|x)0 ) ψS (7)

φ|x)0 ) 0 (8)

1666 Borukhov et al. Macromolecules, Vol. 31, No. 5, 1998

length scales in the problem: (i) the Edwards correlationlength ê ) a/(υφb

2)1/2, characterizing the concentrationfluctuations of neutral polymer solutions; (ii) the De-bye-Huckel screening length κs

-1 ) (8πlBcb)-1/2 wherelB ) e2/εkBT is the Bjerrum length equal to about 7 Åfor aqueous solutions at room temperature. Additionallength scales can be associated with electrostatic and/or short range surface interactions.Motivated by the numerical results (Figure 2), we

assume that the balance between these interactionsresults in one dominant length scale D characterizingthe adsorption at the surface. Hence, we write thepolymer order parameter profile in the form

where h is a dimensionless function normalized to unityat its maximum and cm sets the scale of polymeradsorption. The free energy can be now expressed interms of D and cm, while the exact form of h(z) affectsonly the numerical prefactors.In principle, the adsorption length D depends also on

the ionic strength through κs-1. As discussed below the

scaling assumption (eq 9) is only valid as long as κs-1

and D are not of the same order of magnitude. Other-wise, h should be a function of both κsx and x/D. Weconcentrate now on two limiting regimes where eq 9 canbe justified: (i) the low-salt regime D , κs

-1 and (ii) thehigh-salt regime D . κs

-1.Low-Salt Regime: D , Ks

-1. In the low-salt regimethe effect of the small ions can be neglected and the freeenergy, eqs 1-4, is approximated by (see also ref 26)

The first term is the elastic energy characterizing theresponse of the polymer to concentration inhomogene-ities. The second term accounts for the electrostaticattraction of the polymers to the charged surface. The

third term represents the Coulomb repulsion betweenadsorbed monomers. Indeed, the interaction betweentwo layers with charge densities per unit area σ )peφ2(x)dx and σ′ ) peφ2(x′)dx′, is proportional to theirdistance |x - x′| and yields the D3 dependence. The lastterm represents the excluded volume repulsion betweenadsorbed monomers, where we assume that the mono-mer concentration near the surface is much larger thanthe bulk concentration cm . φb

2. The coefficients A1,A2, B1, and B2 are numerical prefactors, which dependon the exact shape of the dimensionless function h(z).These coefficients can be explicitly calculated for aspecific profile by integrating the Poisson equationwithout taking into account the small ion contribu-tions.31 For a linear profile, h(z) ) z for 0 e z e 1 andh(z) ) 0 for z > 1, we get A1 ) 1, A2 ) 1/3, B1 ) 1/14,and B2 ) 1/5; for a parabolic profile, h(z) ) 4z(1 - z) for0 e z e 1 and h(z) ) 0 for z > 1, we get A1 ) 16/3, A2 )8/15, B1 = 1/9 and B2 = 2/5.In the low-salt regime and for strong enough poly-

electrolytes the electrostatic interactions are muchstronger than the excluded volume interactions. Ne-glecting the latter interactions and minimizing the freeenergy with respect to D and cm gives:

and

As discussed above, these expressions are valid aslong as (i) D , κs

-1 and (ii) the excluded volumeterm in eq 10 is negligible. Condition (i) translates intocb , p|yS|/(8πlBa2). For |yS| = 1, a ) 5 Å and lB ) 7 Å,this limits the salt concentration to cb/p , 0.4 M.Condition (ii) on the magnitude of the excluded volumeterm can be shown to be equivalent to p . v|yS|/lBa2.These requirements are consistent with the data pre-sented in Figure 2.We recall that the profiles presented in Figure 2 were

obtained from the numerical solution of eqs 5 and 6,including the effect of small ions and excluded volume.The scaling relations are verified by plotting in Figure3 the same sets of data as in Figure 2 using rescaledvariables as defined in eqs 11 and 12. Namely, therescaled electrostatic potential ψ(x)/ψS and polymerconcentration c(x)/cm ∼ c(x)a2/|yS|2 are plotted as func-tions of the rescaled distance x/D ∼ xp1/2|yS|1/2/a. Thedifferent curves roughly collapse on the same curve.Note, however, that although the scaling of the spatial

coordinate x with D is satisfactory, the scaling of theconcentration c(x) with cm is not as good, especially atthe peak. Naturally, our simple scaling approach isexpected to lose some of the local details and give abetter description of global properties, and in particularthe amount of polymer drawn to the surface. In manyexperiments the total amount of adsorbed polymer perunit area Γ is measured as a function of the physicalcharacteristics of the system such as the charge fractionp, the pH of the solution or the salt concentration cb.1-8

This quantity can be easily obtained from our scalingexpressions yielding

Figure 2. Adsorption profiles obtained by numerical solutionsof eqs 5 and 6 for several sets of physical parameters in thelow-salt limit. The polymer concentration scaled by its bulkvalue φb

2 is plotted as a function of the distance from thesurface. The different curves correspond to: p ) 1, a ) 5 Å,and yS ) âeψS ) -0.5 (solid curve); p ) 0.1, a ) 5 Å, and yS )-0.5 (dots); p ) 1, a ) 5 Å, and yS ) -1.0 (short dashes); p )1, a ) 10 Å, and yS ) -0.5 (long dashes); and p ) 0.1, a ) 5Å, and yS ) 1.0 (dot-dash line). For all cases φb

2 ) 10-6 Å-3,υ ) 50 Å3, ε ) 80, T ) 300 K, and cb ) 0.1 mM.

φ(x) ) xcmh(x/D) (9)

âF ) A1a2

6Dcm - A2p|yS|cm D +

4πB1lBp2 cm

2D3 + 12B2vcm

2D (10)

D2 ) (5A1

6A2) a2

p|yS|∼ 1p|yS|

(11)

cm ) ( 12A22

25A1B1) |yS|24πlBa

2∼ |yS|2 (12)

Macromolecules, Vol. 31, No. 5, 1998 Scaling Laws of Polyelectrolyte Adsorption 1667

The adsorbed amount Γ(p) in the low-salt regime isplotted in the inset of Figure 4a. As a consequence ofeq 13, Γ decreases with increasing charge fraction p.Similar behavior was also reported in experiments.4This effect is nontrivial, because as the polymer chargeincreases, the chains are subject to a stronger attractionto the surface. On the other hand, the monomer-monomer repulsion is stronger and indeed, in thisregime, the monomer-monomer Coulomb repulsionscales as (pcm)2, and dominates over the adsorptionenergy which scales as pcm.32

High-Salt Regime: D . Ks-1. The opposite case

occurs when D is much larger than κs-1. In this case

the electrostatic interactions are short ranged with acutoff κs

-1. The free energy then reads:

The electrostatic cutoff enters in two places. In thesecond term only the first layer of width κs

-1 interactselectrostatically with the surface. In the third termeach charged layer situated at point x interacts onlywith layers at x′ for which |x - x′| < κs

-1. Since theinteraction between two charged layers at x and x′ isproportional to |x - x′|, the integral over x′ contributesto the κs

-2 dependence, and the integral over x contrib-utes an additional factor of D. This term can be also

Figure 3. Scaling behavior of polyelectrolyte adsorption inthe low-salt regime (eqs 11 and 12). (a) The rescaled electro-static potential ψ(x)/|ψS| as a function of the rescaled distancex/D. (b) The rescaled polymer concentration c(x)/cm as afunction of the same rescaled distance. The profiles are takenfrom Figure 2 (with the same notation). The numericalprefactors of the linear h(x/D) profile were used in thecalculation of D and cm.

Γ ) ∫0∞[c(x) - φb2]dx = Dcm =

|yS|3/2

lBap1/2∼ |yS|

3/2

p1/2(13)

Figure 4. Typical adsorbed amount Γ as a function of (a) thecharge fraction p and (b) the pH - pK0 of the solution fordifferent salt concentrations (eq 17). The insets correspond tothe low-salt regime (eq 13). The parameters used for ε, T, andυ are the same as in Figure 2, and yS ) -0.5 and a ) 5 Å. Thebulk concentration φb

2 is assumed to be much smaller than cm.The numerical prefactors of the linear h(x/D) were used.

âF ) A1a2

6Dcm - A2p|yS|cm κs-1 +

4πB1lBp2κs

-2cm2D + 1

2B2υcm

2D (14)


viewed as an additional electrostatic excluded volumewith υel ∼ lB(p/κs)2. The crude box-like simplificationof screening can be justified when the screening lengthκs

-1 is much smaller than the adsorption length D, andaffects only the numerical factors A2 and B1.Minimization of the free energy gives

and

yielding

The adsorption behavior is depicted in Figures 4 and5. Our results are in agreement with numerical solu-tions of discrete lattice models (the multi-Stern layertheory).9-11,19-23 In Figure 4, Γ is plotted as functionof p (Figure 4a) and the pH (Figure 4b) for different saltconcentrations. The behavior as seen in Figure 4brepresents annealed polyelectrolytes where the nominalcharge fraction is controlled by the pH of the solutionthrough

where pK0 ) -log10K0 and K0 is the apparent dissocia-tion constant.Another observation which can be deduced from eq

17 is that Γ is only a function of p/cb1/2. Indeed, as can

be seen in Figure 4, cb only affects the position of thepeak and not its height.The effect of salt concentration is shown in Figure 5,

where Γ is plotted as function of the salt concentrationcb for two charge fractions p ) 0.01 and 0.25. Thecurves on the right side of the graph are calculated fromthe high-salt expression for Γ (eq 17). The horizontallines on the left side of the graph indicate the low-saltvalues of Γ (eq 13). The dashed lines in the intermediatesalt regime serve only as guides to the eye since ourapproach is not valid when D and κs

-1 are of the sameorder. The behavior in this regime can be deducedqualitatively by comparing the low-salt asymptoticvalues with the high-salt values. The effect of salt inthe intermediate regime depends strongly on the poly-mer charge fraction p. For weak polyelectrolytes (e.g.,p ) 0.01) Γ should vary considerably across the inter-mediate regime, and is expected to decrease as salt isadded to the solution. On the other hand, for strongpolyelectrolytes (e.g., p ) 0.25), the adsorbed amountdoes not change much across this regime.Attention should be drawn to the distinction between

weak and strong polyelectrolytes. For weak polyelec-trolytes, the adsorbed amount Γ is a monotonouslydecreasing function of the salt concentration cb in thewhole range of salt concentrations. The reason beingthat the monomer-monomer Coulomb repulsion, pro-portional to p2, is weaker than the monomer-surfaceinteraction, which is linear in p.For strong polyelectrolytes, on the other hand, the

balance between these two electrostatic terms dependson the amount of salt. At low salt concentrations, thedominant interaction is the monomer-surface Coulombrepulsion. Consequently, addition of salt screens thisinteraction and increases the adsorbed amount. Whenthe salt concentration is high enough, this Coulombrepulsion is screened out and the effect of salt is toweaken the surface attraction. At this point the ad-sorbed amount starts to decrease. As a result, thebehavior over the whole concentration range is non-monotonic with a maximum at some optimal value cb*as seen in Figure 5.From this analysis and from Figures 4 and 5 and eq

17, it is now natural to divide the high-salt regime intotwo sub regimes according to the polyelectrolyte charge.At low charge fractions (subregime HS I), p , p* )(cbυ)1/2, the excluded volume term dominates the de-nominator of eq 17 and

whereas at high p (subregime HS II), p . p*, themonomer-monomer electrostatic repulsion dominatesand Γ decreases with p and increases with cb:

The various regimes with their crossover lines areshown schematically in Figure 6. Keeping the chargefraction p constant while changing the amount of saltcorresponds to a vertical scan through the diagram. Forweak polyelectrolytes this cut goes through the left sideof the diagram starting from the low-salt regime and,upon addition of salt, into the HS I regime. Such a pathdescribes the monotonous behavior inferred from Figure

Figure 5. The adsorbed amount Γ as a function of the saltconcentration cb for p ) 0.01 and 0.25. The solid curves on theright side correspond to the scaling relations in the high-saltregime (eq 17). The horizontal lines on the left side mark thelow-salt values (eq 13). The dashed lines serve as guides tothe eye. The parameters used are ε ) 80, T ) 300 K, υ )50 Å3, a ) 5 Å, and yS ) -2.0 and the numerical prefactors ofthe linear h(x/D).

D ) ( A1

2A2) κsa2p|yS|

∼ cb1/2

p|yS|(15)

cm ∼p2|yS|2/(κsa)2

B1p2/cb + B2v

(16)

Γ ∼ p|yS|cb-1/2

B1p2/cb + B2v

(17)

p ) 10pH-pK0

1 + 10pH-pK0(18)

Γ ∼ p|yS|cb1/2

(19)

Γ ∼ cb1/2 |yS|p

(20)


5 for the weak polyelectrolyte (p ) 0.01). For strongpolyelectrolytes the cut goes through the right side ofthe diagram, starting from the low-salt regime, passingthrough the HS II regime, and ending in the HS Iregime. The passage through the HS II regime isresponsible for the nonmonotonous behavior inferredfrom Figure 5 for the strong polyelectrolyte (p ) 0.25).Similarly, Figures 4a and b correspond to horizontal

scans through the top half of the diagram. As long asthe system is in the HS I regime, the adsorbed amountincreases when the polymer charge fraction increases.As the polymer charge further increases, the systementers the HS II regime and the adsorbed amountdecreases. Thus, the nonmonotonous behavior of Figure4 is exhibited.

IV. Comparison to Experiments

In the previous section the adsorption behavior wasdivided into three distinct regimes (Figure 6): (i) low-salt regime cb , p|yS|/8πlBa2; (ii) first high-salt (HS I)regime, where cb . p|yS|/8πlBa2 and p , p* ) (cbυ)1/2(weak polyelectrolytes); and (iii) second high-salt (HSII) regime, where cb . p|yS|/8πlBa2 and p . p* (strongpolyelectrolytes).In the low salt regime Γ ∼ |yS|3/2/p1/2. In the two HS

regimes Γ scales differently Γ ∼ p|yS|/cb1/2 in the HS Iregime, and Γ ∼ cb

1/2|yS|/p in the HS II regime. In thefollowing we compare our results with experimentalmeasurements done on different types of polyelectro-lytes and using different experimental techniques. Ourscaling results are in good agreement with adsorptionexperiments, although in real systems the chargedistribution of the polyelectrolytes can be more compli-cated than in our simple model.Low-Salt Regime. Denoyel et al.4 have studied the

adsorption of heteropolymers made of neutral (acryla-mide) and cationic monomers (derived from chlorideacrylate). The fractional charge was fixed during thepolymerization process and varied from p ) 0 to 1.

Because the salt amount in their experiment was quitelow, 1.2 mM corresponding to κs

-1 = 90 Å, their experi-mental range satisfies the low-salt conditions. Indeed,the measured Γ (Table 2 in ref 4) exhibits a p-1/2

dependence as in eq 13.Weak Polyelectrolytes: Effect of Salt. Shubin

and Linse7 adsorbed another cationic derivative ofpolyacrylamide on silica. The fractional charge wasfixed at a low value (p ) 0.034), and the salt concentra-tion varied from cb ) 0.1 mM to cb ≈ 0.2 M. Ellipsom-etry was used to measure Γ and D of the adsorbed layeras function of the salt concentration. This low chargefraction belongs to the left side (low p) of Figure 6. Theexperimental behavior is similar to our predictions asshown in Figure 5 for weak polyelectrolytes. At lowelectrolyte concentration (cb < 1 mM), the adsorbedamount is essentially constant and decreases at highersalt concentration (HS I regime of Figure 6). Similarbehavior was obtained both by numerical calculationsusing the multi-Stern layer model,7,22,23 and in otheradsorption experiments of cationic potato starch.6

Strong Polyelectrolytes: Effect of Salt. Kawagu-chi et al.2 measured the adsorption of a highly chargedpolyelectrolyte poly(4-vinyl-N-n-propylpyridinium bro-mide) (PVPP) on silica surfaces. Because of the highionic strength this system belongs to the HS II regime.Indeed, Γ scales as cb1/2 was in agreement with ourprediction. Meadows et al.3 also performed adsorptionexperiments with highly charged (p ) 0.9) hydrolyzedpolyacrylamide. The adsorbed amount Γ and the widthof the adsorbed layer D were found to increase uponaddition of salt. Qualitatively, this agrees with ourprediction in the HS II regime. However, the measuredpower laws are weaker than our predictions. A simplepower law fit of their salt dependence gives Γ ∼ cb

1/4 ascompared to our cb

1/2 prediction. This behavior is in-termediate between the salt free and HS II regimes.Effect of Charge Fraction. Peyser and Ullman1

studied the adsorption of poly-4-vinylpyridine (PVP) ona glass surface as a function of the charge fraction forthree different salt concentrations. The system belongsto the right side (p j 1) of Figure 6 between the low-salt and HS II regimes. As expected Γ increases withcb and decreases with p. Moreover, it is possible to fitthe data to a simple scaling law of the form Γ ∼cb1/4/p1/2. Our scaling results do not fit very well theseexperiments which lie in the intermediate regime,between the low-salt and HS II regimes.In experiments on annealed polyelectrolytes,5,8 the

polymer charge can be tuned by the pH of the solution(eq 18). The behavior then shifts continuously from theHS I to the HS II regimes. For example, Blaakmeer etal.5 used polyacrylic acid which is neutral (no dissocia-tion) at low pH but becomes negatively charged (strongdissociation) at higher pH. As predicted by eq 17 (seealso Figure 4b), a nonmonotonous dependence of Γ onthe pH was observed, with a maximum below the pK0.This effect had been already verified by numericalcalculations based on the multi-Stern layer model.5

A similar maximum in Γ was also observed in adsorp-tion experiments of proteins12 and diblock copolymerswith varying ratios between the charged and neutralblocks8 and may be interpreted using similar consider-ations.

Figure 6. Schematic diagram of the different adsorptionregimes as a function of the charge fraction p and thesalt concentration cb. Three regimes can be distinguished:(i) the low-salt regime D , κs

-1; (ii) the high-salt regime(HS I) D . κs

-1 for weak polyelectrolytes p , p* ) (cbυ)1/2; and(iii) the high-salt regime (HS II) D . κs

-1 for strong polyelec-trolytes p . p*.


V. Conclusions

In this work we used simple arguments to derivescaling laws describing the adsorption of polyelectrolyteson a single charged surface held at a constant potential.We obtained expressions for the amount of adsorbedpolymer Γ and the width D of the adsorbed layer, as afunction of the fractional charge p and the salt concen-tration cb. In the low-salt regime, a p-1/2 dependenceof Γ is found. It is supported by our numerical solutionsof the profile, eqs 5 and 6, and is in agreement with theexperiment.4 This behavior is due to strong Coulombrepulsion between adsorbed monomers in the absenceof salt. As p decreases, the adsorbed amount increasesuntil the electrostatic attraction becomes weaker thanthe excluded volume repulsion, at which point Γ startsto decrease rapidly. At high salt concentrations weobtain two limiting behaviors: (i) for weak polyelectro-lytes, p , p* ) (cbυ)1/2, the adsorbed amount increaseswith the fractional charge and decreases with the saltconcentration, Γ ∼ p/cb1/2, because of the monomer-surface electrostatic attraction. (ii) For strong polyelec-trolytes, p . p*, the adsorbed amount decreases withthe fractional charge and increases with the salt con-centration, Γ ∼ cb1/2/p, because of the dominance of themonomer-monomer electrostatic repulsion. Betweenthese two regimes we find that the adsorbed amountreaches a maximum in agreement with other experi-ments.5,8

The scaling approach can serve as a starting pointfor further investigations. Special attention should bedirected to the crossover regime where D and κs

-1 are ofcomparable size. At present, it is not clear whether theintermediate regime represents simply a crossoverbetween regimes or is a scaling regime on its own.Another important question addresses the relativeimportance of attractive versus repulsive forces betweentwo charged surfaces in the presence of a polyelectrolytesolution. Finally, our approach could be used in non-planar geometries such as spheres (colloidal particles)and cylinders.

Acknowledgment. We would like to thank L. Au-vray, M. Cohen-Stuart, J. Daillant, H. Diamant, P.Guenoun, Y. Kantor, P. Linse, P. Pincus, S. Safran andC. Williams for useful discussions. Two of us (I.B. andD.A.) would like to thank the Service de PhysiqueTheorique (CE-Saclay) and one of us (H.O.) would liketo thank the Sackler Institute of Solid State Physics (TelAviv University) for their hospitality. Partial supportfrom the German-Israel Foundation (G.I.F) under grantno. I-0197 and the U.S.-Israel Binational Foundation(B.S.F.) under grant No. 94-00291 is gratefully acknowl-edged.

References and Notes

(1) Peyser, P.; Ullman, R. J. Polym. Sci. A 1965, 3, 3165.(2) Kawaguchi, M.; Kawaguchi, H.; Takahashi, A. J. Colloid

Interface Sci. 1988, 124, 57.

(3) Meadows, J.; Williams, P. A.; Garvey, M. J.; Harrop, R.;Phillips, G. O. J. Colloid Interface Sci. 1989, 132, 319.

(4) Denoyel, R.; Durand, G.; Lafuma, F.; Audbert, R. J. ColloidInterface Sci. 1990, 139, 281.

(5) Blaakmeer, J.; Bohmer, M. R.; Cohen Stuart, M. A.; Fleer.G. J. Macromolecules 1990, 23, 2301.

(6) Van de Steeg. H. G. M.; de Keizer, A.; Cohen Stuart, M. A.;Bijsterbosch, B. H. Colloid Surf. A 1993, 70, 91.

(7) Shubin, V.; Linse, P. J. Phys. Chem. 1995, 99, 1285.(8) Hoogeveen, N. G. Ph.D. Thesis, Wageningen Agricultural

University, The Netherlands, 1996.(9) Cohen Stuart, M. A. J. Phys. France 1988, 49, 1001.(10) Cohen Stuart, M. A.; Fleer, G. J.; Lyklema, J.; Norde, W.;

Scheutjens, J. M. H. M. Adv. Colloid Interface Sci. 1991, 34,477.

(11) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.;Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman& Hall: London, 1993; chapter 11.

(12) Haynes, C. A.; Norde, W. Colloid Surf. B 1994, 2, 517.(13) Auroy, P.; Auvray, L.; Leger, L. Macromolecules 1991, 24,

2523.(14) Guiselin, O.; Lee, L. T.; Farnoux, B.; Lapp, A. J. Chem. Phys.

1991, 95, 4632.(15) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971.(16) De Gennes, P. G. Scaling Concepts in Polymer Physics;

Cornell Univ.: Ithaca, 1979.(17) Odijk, T. Macromolecules 1979, 12, 688.(18) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M.Macromolecules

1995, 28, 1859.(19) Van der Schee, H. A.; Lyklema, J. J. Phys. Chem. 1984, 88,

6661.(20) Papenhuijzen, J.; Van der Schee, H. A.; Fleer, G. J. J. Colloid

Interface Sci. 1985, 104, 540.(21) Evers, O. A.; Fleer, G. J.; Scheutjens, J. M. H. M.; Lyklema,

J. J. Colloid Interface Sci. 1985, 111, 446.(22) Van de Steeg, H. G. M.; Cohen Stuart, M. A.; de Keizer, A.;

Bijsterbosch, B. H. Langmuir 1992, 8, 8.(23) Linse, P. Macromolecules 1996, 29, 326.(24) Muthukumar, M. J. Chem. Phys. 1987, 86, 7230.(25) Varoqui, R.; Johner, A.; Elaissari, A. J. Chem. Phys. 1991,

94, 6873.(26) Varoqui, R. J. Phys. II 1993, 3, 1097.(27) Borukhov, I.; Andelman, D.; Orland, H. Europhys. Lett. 1995,

32, 499; Borukhov, I.; Andelman, D.; Orland, H. In Short andLong Chains at Interfaces; Daillant, J., Guenoun, P., Marques,C., Muller, P., Tran Thanh Van, J., Eds.; Edition Frontieres:Gif-sur-Yvette, 1995; pp. 13-20.

(28) Borukhov, I.; Andelman, D.; Orland, H. Eur. Phys. J. D,submitted for publication.

(29) If a constant surface charge density σ is assumed, theboundary condition then becomes ψ′|S ) -4πσ/ε. In realsystems, some of the ionized surface sites can be neutralizedby the binding of small ions from the solution. The actualsurface charge is not fixed but depends on the surfacepotential (see also ref. 30). Neither the surface potential northe surface charge are fixed, and the boundary conditions aremixed. The choice of simple boundary conditions is only anapproximation of a real surface-solution system. The qualityof the approximation depends on the details of the experi-mental system.

(30) For a review see: Israelachvili, J. N. Intermolecular andSurface Forces, 2nd ed.; Academic Press: London, 1990.

(31) For a given profile h(x/D), the electrostatic potential, whichis also a function of x/D, can be calculated by integrating thecharge density from the surface up to a distance x. Thesurface charge density is determined from charge neutralitywhich is imposed by setting the electric field to zero at x )D. The electrostatic term in the free energy fel is thenobtained by integrating pecm h2(x/D)ψ(x).

(32) This behavior is valid for a wide range of p values. However,as p f 0, eventually the excluded volume interaction will takeover and Γ will start to decrease to zero.

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